Problem: What is the slope of the line tangent to $f(x) = -x^{2}-x+7$ at $x = -3$ ?
The slope of the tangent line is $ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ $ = \lim_{h \to 0} \frac{(-(x+h)^{2}-(x+h)+7) - (-x^{2}-x+7)}{h}$ $ = \lim_{h \to 0} \frac{(-(x^{2}+2x h+h^{2})-(x+h)+7) - (-x^{2}-x+7)}{h}$ $ = \lim_{h \to 0} \frac{-x^{2}-2(x h)-h^{2}-x-h+7+x^{2}+x-7}{h}$ $ = \lim_{h \to 0} \frac{-2(x h)-h^{2}-h}{h}$ $ = \lim_{h \to 0} -2x-h-1$ $ = -2x-1$ $ = (-2)(-3)-1$ $ = 5$